Bulletin of the American Physical Society
60th Annual Meeting of the Divison of Fluid Dynamics
Volume 52, Number 12
Sunday–Tuesday, November 18–20, 2007; Salt Lake City, Utah
Session AK: Rarefied Gases and DSMC |
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Chair: M. Gallis, Sandia National Laboratories Room: Salt Palace Convention Center 250 E |
Sunday, November 18, 2007 8:30AM - 8:43AM |
AK.00001: Macroscopic equations for rarefied gases from the elimination of fast variables Paul Dellar We derive macroscopic descriptions of rarefied gases from a hierarchy of moment equations (Maxwell's equation of transfer) using van Kampen's procedure for eliminating fast variables. The elimination leaves closed evolution equations for the five moments unaffected by collisions -- the mass, momentum, and energy densities. We show that the equations of Chen, Rao, and Spiegel [2000, \textit{Phys. Lett. A}, \textbf{271}, 87], like the Navier--Stokes--Fourier equations, emerge from van Kampen's procedure. We propose related equations using the concept of a translational temperature, following work on polyatomic gases. These equations offers excellent agreement with experimental data on the phase speed of ultrasound, in both the continuum and highly rarefied limits. They are equivalent to a nonlocal relation for the entropy production rate, with a sampling distance of a mean free path. Using moment equations offers a definitive treatment of the Prandtl number problem using model collision operators, greatly reduces the labor of deriving equations for different collision operators, and clarifies the r\^ole of solvability conditions applied to the distribution function. [Preview Abstract] |
Sunday, November 18, 2007 8:43AM - 8:56AM |
AK.00002: Lattice Boltzmann Computations of Micro Channel Gas Flows Using Different Boundary Conditions Frank Chambers, Taiho Yeom Lattice Boltzmann method simulations were performed for isothermal gas flows through a micro channel at Knudsen (Kn) numbers of 0.00194, 0.0194, and 0.194 and Reynolds numbers between 2 and 12. The goal of the work was to compare the performance of the no-slip bounce back, the accommodation coefficient, and the reflection factor boundary condition schemes, particularly for slip flows. For the micro channel flows considered, the reflection factor boundary condition provided results which matched best with the literature, while the no-slip bounce-back and accommodation coefficient boundary conditions showed greater deviations. For Kn = 0.0194, a reflection factor of 0.85 yielded velocity profiles in very good agreement with the unified flow analytical results with deviations of approximately 0.3\% in the maximum velocity and 2.3\% in the slip velocity. For Kn = 0.00194, the no-slip bounce- back and 0.85 reflection factor yield good velocity profiles although there are deviations from the analytical solution near the wall. However, in the transitional flow regime at Kn = 0.194, none of the approaches yielded acceptable results. [Preview Abstract] |
Sunday, November 18, 2007 8:56AM - 9:09AM |
AK.00003: Accuracy of higher-order lattice Boltzmann methods for micro-scale flows with finite Knudsen numbers Seung Hyun Kim, Heinz Pitsch, Iain Boyd The accuracy of the lattice Boltzmann (LB) method for micro-scale flows with finite Knudsen numbers is investigated. We employ up to eleventh-order Gauss-Hermite quadrature and a diffuse-scattering boundary condition for fluid-wall interactions. Detailed comparisons with the Direct Simulation Monte Carlo (DSMC) method and the linearized Boltzmann equation are made for planar Couette and Poiseuille flows. With a consistent definition of the Knudsen number, the slip coefficients of the LB equation with the standard D2Q9 scheme are found to be slightly larger than those of the Boltzmann equation with the same boundary condition, which makes the standard LB method remain quantitatively accurate only for small Knudsen numbers. While all higher-order LB methods considered here perform better than the standard LB method, accuracy of the LB hierarchy does not monotonically increase with the order of the Gauss-Hermite quadrature. [Preview Abstract] |
Sunday, November 18, 2007 9:09AM - 9:22AM |
AK.00004: A low-variance deviational simulation Monte Carlo for the Boltzmann equation Nicolas Hadjiconstantinou, Thomas Homolle We present an efficient particle method for solving the Boltzmann equation. The key ingredients of this work are the variance reduction ideas presented in [L. L. Baker and N. G. Hadjiconstantinou, Physics of Fluids, vol. 17, art. no 051703, 2005] and a new collision-integral formulation which allows the method to retain the algorithmic structure of direct simulation Monte Carlo (DSMC) and thus enjoy the numerous advantages associated with particle methods, such as a physically intuitive formulation, computational efficiency due to importance sampling, low memory usage (no discretization in velocity space), and the ability to naturally and accurately capture discontinuities in the distribution function. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage for low signal flows (e.g. low flow speed) compared to traditional particle methods such as DSMC. In particular, the resulting method can capture arbitrarily small deviations from equilibrium at a computational cost that is independent of the magnitude of this deviation. The method is validated by comparing its predictions with DSMC solutions for spatially homogeneous and inhomogeneous problems. [Preview Abstract] |
Sunday, November 18, 2007 9:22AM - 9:35AM |
AK.00005: Convergence Behavior of Bird's Sophisticated DSMC Algorithm M.A. Gallis, J.R. Torczynski, D.J. Rader Bird's standard Direct Simulation Monte Carlo (DSMC) algorithm has remained almost unchanged since the mid-1970s. Recently, Bird developed a new DSMC algorithm, termed ``sophisticated DSMC'', which significantly modifies the way molecules both move and collide. The sophisticated DSMC algorithm is implemented in a one-dimensional DSMC code, and its convergence behavior is investigated for one-dimensional Fourier flow, where an argon-like hard-sphere gas is confined between two parallel, motionless, fully accommodating walls with unequal temperatures. As in previous work, the primary convergence metric is the ratio of the DSMC-calculated thermal conductivity to the theoretical value. The convergence behavior of sophisticated DSMC is compared to that of standard DSMC and to the predictions of Green-Kubo theory. The sophisticated algorithm significantly reduces the computational resources needed to maintain a fixed level of accuracy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. [Preview Abstract] |
Sunday, November 18, 2007 9:35AM - 9:48AM |
AK.00006: Evaluation of Uncertainty Evolution in Initial Conditions by the Least Square Kernel Density Function Method Carlos Pantano, Babak Shotorban We present an approximation method to solve the probability density function (pdf) in the Liouville equation encountered in the evolution of uncertainty of initial values in dynamical systems. A state-space based method is implemented through a least-squares projection using global functional approximations. The analytical elementary (kernel density) functions have parameters whose temporal evolution is obtained by the present method. The realizability conditions of the probability density, normalization and non-negativity, are enforced at all times. The method is successfully tested for the evolution of uncertainty in a Riccati equation and in a particle moving in a one-dimensional fluid under the influence of Stokes drag force. Predicted results compare well against the results obtained by Monte-Carlo Simulations. [Preview Abstract] |
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